VIEWS: 7 PAGES: 25 CATEGORY: Technology POSTED ON: 4/17/2010
Analyzing Queuing Systems with Coupled Processors through Semideﬁnite Programming Balaji Rengarajan, Constantine Caramanis, and Gustavo de Veciana Dept. of Electrical and Computer Engineering The University of Texas at Austin {balaji, cmcaram, gustavo}@ece.utexas.edu June 30, 2008 Abstract We consider queuing systems with coupled processors, where the service rate at each queue varies depending on the set of queues in the system with non-zero queue lengths. In general, such queuing systems are very difﬁcult to analyze and steady state queue length distributions are known only for two-queue systems. The coupled-processors model arises naturally in the study of several systems where a resource is shared by several classes of customers. We study the stochastic recursive equations that govern such systems, and obtain lower and upper bounds on the moments of the queue length by formulating a moments prob- lem and solving a semideﬁnite relaxation of the original problem. The resulting bounds are very close even at lower-order relaxations, and can be made progressively tighter at the ex- pense of increasing the complexity of the associated semideﬁnite program. We demonstrate the effectiveness of our proposed methodology in studying and optimizing dynamic systems such as interference-coupled wireless networks and GPS servers. 1 Introduction We study a queuing system with coupled processors, where the rate at which users in a queue are served depends on the lengths of the other queues in the system. In particular, we consider systems where the service rate at each queue varies depending on the set of queues in the system 1 with non-zero queue lengths. This coupled-processors model arises naturally in the study of systems where a resource is shared by several classes of customers, and has been studied for a long time in the queuing literature in the context of bandwidth sharing of elastic ﬂows in packet networks. Generalized Processor Sharing (GPS) [1] is a service discipline that was developed to allow capacity to be shared in a fair and ﬂexible manner, and is an important mechanism for achieving differentiated quality of service. Unlike a strict priority scheme, the GPS discipline allows for service differentiation while preventing starvation. The following is a brief description of the GPS mechanism. Consider N classes of customers, each with some associated arrival rate and service requirements, sharing a server. Each class, n, also has an associated weight φn , with ∑N φn = 1 and can be served at rate Rn . If all the queues are non-empty, the fraction of time n=1 queue n is served is given by φn and the corresponding service rate is φn Rn . However, if some of the queues are empty, their allotted time-fractions are distributed among the non-empty queues in proportion to their respective weights. Thus, the queues are coupled by the mechanism used to share excess capacity. While the GPS policy is not itself realistic to implement, disciplines that closely track the GPS mechanism such as Weighted Fair queuing (WFQ) [2] are used in practice. Understanding the performance of the system, and of the user classes is very important, for example, in order to optimize the choice of the class weights. Another area where the coupled-processors model can be applied is in the study of interference- limited wireless networks. As network deployments with increased base station/access point den- sities are used to meet ever increasing demands for capacity, wireless systems are forced to oper- ate in a highly dynamic, interference limited regime. There is a huge disparity in the data rates that interference-limited users perceive when neighboring access points/base stations are idle and when there are concurrent interfering transmissions in neighboring cells. However, wireless net- works have mostly been studied assuming that they serve static user populations (a ﬁxed set of inﬁnitely backlogged users), and that adjacent transmitters are always interfering with each other. In a more realistic scenario, data requests from users are generated at random times, and the users leave when their service requirements have been met. Thus, it is not the case that cells always have active users; in fact base stations will be idle for some load-dependent fraction of time. Inter-cell 2 interference couples the service rates that users in different cells perceive, and thus their perfor- mance. This dynamic scenario has been examined in [3, 4, 5], and it was demonstrated that the performance perceived by users is very different from that predicted by the analysis of the static system. In this paper, we will consider (brieﬂy) the effect of time-varying interference on delays experienced by best-effort downlink users. A further example for coupling in wireless networks, that we will study is the performance of a simple contention-based multiple-access scheme. In general, coupled queuing systems are very difﬁcult to analyze. Large deviations asymp- totics of the workload in GPS systems have been derived in [6, 7, 8, 9], and the effect of serving customers with long-tailed service times has been analyzed in [10, 11]. However, steady state queue length distributions are known only in some special cases. The coupled-processors model with exponentially distributed service requests and the number of classes/queues restricted to 2 was analyzed in [12, 13, 14]. The uniformization technique was used in [13] to study the model with φ1 = φ2 and R1 = R2 . The generating function of the steady-state queue length distribution was obtained in [12] by solving a Riemann-Hilbert boundary value problem, and closed-form ex- pressions were obtained in [14] for the work-conserving case where R1 = R2 , without resorting to the formulation of a Riemann-Hilbert problem. However, the techniques used in the above papers cannot be extended to systems with more than two queues. In this paper, we obtain bounds on the moments of the steady-state queue-length in a system consisting of N coupled queues by studying the stochastic recursive equations that govern the system. We obtain lower and upper bounds on the moments of the queue length by formulating and solving a semideﬁnite relaxation of the original problem. These bounds can be made progres- sively tighter at the expense of increasing the complexity of the associated semideﬁnite program. Our model is motivated by the one used in [15] to analyze GI/GI queues, and draws on results obtained in [16, 17]. The derivation in [15] is unable to accommodate state-dependent service times and coupled processors. A main contribution of this paper is to extend the moment-based SDP approach to this broader class of problems. We then consider some example applications, to illustrate the importance of being able to accurately capture the impact of coupling. The rest of the paper is organized as follows: The system model is described in Section 2, and the stochastic recursive equation governing the system is described. In Section 3, the 3 moments based approach to bounding the functions of interest is constructed, and the semideﬁnite relaxation of the moment problem is derived. We apply our proposed methodology to model, and optimize some coupled systems in Section 4. Section 4.1 considers the example of interference- limited wireless networks, Section 4.2 illustrates the problem of studying and optimizing GPS servers, and Section 4.3 examines the design of a multiple access system. Finally, Section 5 concludes the paper. 2 System Model and Notation We consider a system of N queues. Users arrive at queue n as a Poisson process, with mean arrival rate λn . We assume that the users require exponential service times with mean 1. The users are served according to a ﬁrst come ﬁrst serve (FCFS) policy at each queue. We denote the queue length at each queue at time t by Qn (t), n = 1, 2, . . . , N. The service rates at the queues depends on the subset of queues in the system with non-zero queue length, where the number of possible subsets is 2N . The status(busy or idle) of the queues in the system is captured by a vector ∆(t) of length n that takes values δi , i = 1, . . . , 2N . The nth component of ∆(t), ∆n (t) is equal to 0 when Qn (t) = 0, and 1 when Qn (t) > 0. The rate at which queue n serves users when the state of the ∆(t) queues is ∆(t) is denoted by µn . We assume that the queuing system is stable, and also that there is a well deﬁned maximum rate µ∗ that bounds the rate at which any queue can be served, irrespective of the state of the system. The queue length process evolves as a continuous time Markov chain, parametrized by the rates deﬁned above. In this paper, we will study the embedded discrete time Markov chain ob- tained after uniformizing this continuous time Markov chain. Since the maximum rate of transi- tions is bounded by M = ∑N λn + Nµ∗ , the continuous time Markov chain can be uniformized by n=1 introducing ﬁctitious events that cause no change in the state of the Markov chain. We denote the state of the uniformized discrete time Markov chain at step k by Q(k) = {Qn (k), n = 1, 2, . . . , N}, and analogously, the state of the queues by ∆(k). The transition probabilities for the uniformized 4 Markov chain when Qn (k) = qn are as follows: λn Pr.(Arrival into queue n) = , n = 1, . . . , N M ∆(k) µn Pr.(Departure from queue n) = 1(qn > 0), n = 1, . . . , N M ∆(k) ∑N (λn + µn 1(qn > 0)) n=1 Pr.(No change in state) = 1 − M The steady state queue length distribution of this uniformized Markov chain is identical to that of the original. The evolution of the uniformized Markov chain can be expressed in the form of the following stochastic recursive model, where X(k) = {Xn (k), n = 1, 2, . . . , N} denotes the incremental change in the queue lengths that results in the Markov chain transitioning from Q(k) to Q(k + 1). Q(k + 1) = Q(k) + X(k), k = 0, 1, . . . (1) An arrival into queue n at iteration k is represented by Xn (k) = 1, a departure by Xn (k) = −1 and if the transition corresponds to the self-loop, X(k) = 0. Note that X(k) and Q(k) are not independent. The distribution of X(k) clearly depends on ∆(k), and thus on Q(k). When the system is stable, and steady state exists, the stochastic recursive model converges [18, 19], and there exists a joint distribution for the steady state vectors (Q, X) such that d Q = g(Q, X) = Q + X. (2) The equality here is in distribution. Our goal is to characterize the behavior of the queuing system by formulating a moments based approach to bound functions of the moments of Q. As in [15], we only use information on the moments of X derived from the uniformized Markov chain in the formulation. 3 Problem Formulation We denote the steady state marginal probability measure of Q by ψQ and that of X by ψX . These measures are supported on SQ ⊆ RN and SX ⊆ RN respectively. Let ψ denote the joint probability 5 measure for (Q, X), supported on S = SQ × SX . Note that, ψQ and ψX are not necessarily inde- pendent and the joint distribution ψ cannot be expressed as a product form. Using this notation, the distribution of g(Q, X) on the right hand side of Eq. (2) can be written as ψg−1 , and the steady state equation can be represented as ψQ = ψg−1 , (3) where g(Q, X) = Q + X. We partition SQ into regions Sδi , such that the subset of queues with non-zero queue length is identical at all states within each region, i.e., if Q ∈ Sδi , then ∆ = δi . Let ψδi , ψδi , ψδi be Q X the conditional probability measures on Q, X, and (Q, X) respectively, conditioned on Q ∈ Sδi . Note that, given the status vector ∆, the service rates at the various queues are determined. Since specifying the region Sδi that Q belongs to ﬁxes ∆, the distributions ψδi and ψδi are conditionally Q X independent, and we can write ψδi = ψδi ψδi . Q X (4) We use the multi-index notation in the formulation of the optimization problem used to deter- mine the upper and lower bounds. If α = {αn , n = 1, . . . , N} ∈ NN is a N-dimensional multi-index, and Y = {Yn , n = 1, . . . , N} ∈ RN a N-dimensional vector, we denote by Y α the term Y1α1 . . .YN N , α and we let |α| = ∑N αn . We will use information about the conditional moments Eψδi X β , n=1 X β |β| ≤ 2r and i = 1, . . . , 2N , denoted mδi . These conditional moments can be computed in a straight- forward manner as the transition probabilities of the uniformized Markov chain do not change within regions Sδi . We then determine bounds on functions of the form EψQ ∑|γ|≤2r wγ Qγ , where wγ are constant weights. The problem formulation follows: P ROBLEM 3.1 Given Borel measurable sets SQ ⊆ RN and SX ⊆ RN , solve: sup / inf EψQ ∑ wγ Qγ (5) ψ |γ|≤2r s.t. ψQ = ψg−1 (6) ψδi = ψδi ψδi , i = 1, . . . , 2N Q X (7) 6 Eψδi X β = mδi , i = 1, . . . , 2N , |β| ≤ 2r β (8) X Eψ [1] = EψQ [1] = EψX [1] = 1 (9) ψ ∈ M(S ), ψQ ∈ M(SQ ), ψX ∈ M(SX ) (10) Here, M(S ), M(SQ ) and M(SX ) are the sets of ﬁnite positive Borel measures supported by S , SQ and SX respectively. Additionally, the constraints in Eq. (9) guarantee that the measures are valid probability measures. If we let r → ∞, all the moments of X would be speciﬁed, thus specifying exactly the distribution of X which in turn uniquely determines the distribution of Q, and the joint distribution of (Q, X). In this case, the bounds would converge to the exact solution. 3.1 The Moments Based Approach We derive a moments-based relaxation for Problem 3.1, based on joint moments of degree no higher than 2r. The decision variables for this problem are deﬁned as follows: xk := E[Qα X β |Q ∈ Sδk ]Pr.[Q ∈ Sδk ], ∀|α| + |β| ≤ 2r, k = 1, . . . , 2N = K αβ Now, we can express the objective function and the relaxed versions of the constraints in Problem 3.1 in terms of the above decision variables as described below: 3.1.1 Objective Function The objective function in Eq. (5) can be written in terms of the conditional expectations as follows using the theorem of total expectation. This expression can then be expressed as a linear function of the decision variables. K EψQ ∑ wγ Qγ = ∑E ∑ wγ Qγ |Q ∈ Sδk Pr.[Q ∈ Sδk ] |γ|≤2r k=1 |γ|≤2r K γ,0 = ∑ ∑ wγ xk k=1 |γ|≤2r 7 3.1.2 Constraints The equality in distribution in the steady state constraint from Eq. (6) implies that the equality also holds for moments of all orders. We relax this constraint to the following constraint relating joint moments of order less than or equal to 2r: d Q = g(Q, X) = Q + X ⇒ E[Qα ] = E[(Q + X)α ], ∀|α| ≤ 2r. We can then break down the above equation in terms of the conditional expectations, again using the theorem of total expectation to get K K ∑ E[Qα |Q ∈ Sδ ]Pr.[Q ∈ Sδ ] = ∑ E[g(Q, X)α |Q ∈ Sδ ]Pr.[Q ∈ Sδ ], ∀|α| ≤ 2r. k k k k k=1 k=1 The term g(Q, X)α can be expanded using the binomial theorem as (γ ,γ2 ) g(Q, X)α = (Q + X)α = ∑ gα 1 Qγ1 X γ2 , |γ1 |+|γ2 |≤α (γ ,γ2 ) where {gα 1 } are the coefﬁcients resulting from the expansion. The constraint can now be written as a linear function of the decision variables as follows: K K (γ ,γ2 ) ∑ E[Qα |Q ∈ Sδ ]Pr.[Q ∈ Sδ ] k k = ∑E ∑ gα 1 Qγ1 X γ2 |Q ∈ Sδk Pr.[Q ∈ Sδk ] k=1 k=1 |γ1 |+|γ2 |≤α , ∀|α| ≤ 2r K K α,0 (γ ,γ2 ) γ1 γ1 ⇒ ∑ xk = ∑ ∑ gα 1 xk , ∀|α| ≤ 2r. k=1 k=1 |γ1 |+|γ2 |≤α The conditional independence of Q and X in constraint (7) is relaxed by equating the moments of the product of the random variables to the product of the moments. ψδi = ψδi ψδi ⇒ E[Qα X β |Q ∈ Sδk ] = E[Qα |Q ∈ Sδk ]E[X β |Q ∈ Sδk ], ∀|α| + |β| ≤ 2r, k = 1, . . . , K. Q X This along with the given moments of X from constraint (8) can be used to constrain the moments 8 of the joint conditional distribution as follows: E[Qα X β |Q ∈ Sδk ] = E[Qα |Q ∈ Sδk ]E[X β |Q ∈ Sδk ] , ∀|α| + |β| ≤ 2r, k = 1, . . . , K ⇒ E[Qα X β |Q ∈ Sδk ]Pr.[Q ∈ Sδk ] = E[Qα |Q ∈ Sδk ]E[X β |Q ∈ Sδk ]Pr.[Q ∈ Sδk ] , ∀|α| + |β| ≤ 2r, k = 1, . . . , K β ⇒ E[Qα X β |Q ∈ Sδk ]Pr.[Q ∈ Sδk ] = E[Qα |Q ∈ Sδk ]mδk Pr.[Q ∈ Sδk ] , ∀|α| + |β| ≤ 2r, k = 1, . . . , K. Rewriting the above constraint in terms of the decision variables, we get the following set of linear constraints: αβ βα,0 xk = mδk xk , ∀|α| + |β| ≤ 2r, k = 1, . . . , K. In order to ensure that constraint (9) is satisﬁed, and that the joint measure is indeed, a prob- ability measure, we need K ∑ Pr.[Q ∈ Sδ ] k = 1 k=1 K 0,0 ⇒ ∑ xk = 1. k=1 αβ Finally, we need to ensure that constraint (10) is satisﬁed and {xk , |α| + |β| ≤ 2r} represents a valid moment sequence for any k = 1, . . . , K. Partitioning SQ into regions Sδk induces a partition of S = SQ × SX into the regions Sk = Sδk × SX . We denote the cone of moments supported on Sk by: α M2r (Sk ) = xk |xk β = Eψk [Qα X β ], ∀|α| + |β| ≤ 2r and for some ψk ∈ M(Sk ) . Letting M2r (Sk ) denotes the closure of the cone of moments, then, for constraint (10) to be true, {xk } ∈ M2r (Sk ), ∀k = 1, . . . , K. Now, the relaxed version of Problem 3.1 can be expressed in the form of the following conic 9 optimization problem based on joint moments of degree less than or equal to 2r. P ROBLEM 3.2 K γ,0 sup / inf ∑ ∑ wγ xk xk k=1 |γ|≤2r s.t. K K α,0 (γ ,γ2 ) γ1 γ1 ∑ xk = ∑ ∑ gα 1 xk , ∀|α| ≤ 2r k=1 k=1 |γ1 |+|γ2 |≤α αβ βα,0 xk = mδk xk , ∀|α| + |β| ≤ 2r, k = 1, . . . , K K 0,0 ∑ xk = 1 k=1 {xk } ∈ M2r (Sk ), ∀k = 1, . . . , K. 3.2 A Semideﬁnite Relaxation The moment cone can be characterized using positive semideﬁnite matrices as in [15, 17, 20]. A necessary condition for the sequence y = {yαβ , |α| + |β| ≤ 2r} to be a valid truncated sequence of moments is that the associated moment matrix is positive semideﬁnite [17]. Let the moment matrix corresponding to the sequence y be denoted Mr (y). The moment matrix Mr (y) is the i, j block matrix {Mr (y), 0 ≤ i, j ≤ r} with rows and columns indexed in the basis of polynomials of degree less than or equal to r. The entries of the moment matrix satisfy the following condition: if Mr j (y) = yα1 β1 and Mr (y) = yα2 β2 , then Mr j (y) = y(α1 +α1 )(β1 +β2 ) . 1, i,1 i, A necessary condition for y to be a truncated moment sequence is Mr (y) 0 In our formulation, this condition has to be satisﬁed by the decision variables {xk }, ∀k = 1, . . . , K. Furthermore, the regions Sδk can be deﬁned as the intersection of a ﬁnite set of linear in- equalities. If the nth queue is constrained to be non-empty in the region Sδk , then the condition 10 Qn − 1 ≥ 0 holds in Sδk , and in the region Sk . Thus, Sk can be deﬁned by: Sk = ∩θ∈Sk θ(Q, X), where θ is a polynomial of degree 1. Let the coefﬁcients of θ be given by {θα , |α| ≤ 1}. Consider any region Sk , a localizing matrix Mr−1 (θ, xk ) associated with one of its polynomials θ ∈ Sk is deﬁned as: (i, j) η(i, j)+γ Mr−1 (θ, xk ) = ∑ θα xk , |α|≤1 i, j where η(i, j) is the subscript of the entry Mr−1 (xk ) in matrix Mr−1 (xk ). An additional condition which is necessary for the sequence xk to be a valid moment sequence is that the localizing matrix corresponding to each θ ∈ Sk be positive deﬁnite, i.e. Mr−1 (θ, xk ) 0. Relaxing the constraint {xk } ∈ M2r (Sk ), ∀k = 1, . . . , K in Problem 3.2 using the positive semideﬁniteness constraints above, the semideﬁnite relaxation of the moment problem can be written as follows. P ROBLEM 3.3 K γ,0 sup / inf ∑ ∑ wγ xk xk k=1 |γ|≤2r s.t. K K α,0 (γ ,γ2 ) γ1 γ1 ∑ xk = ∑ ∑ gα 1 xk , ∀|α| ≤ 2r k=1 k=1 |γ1 |+|γ2 |≤α αβ α,0 β xk = mδk xk , ∀|α| + |β| ≤ 2r, k = 1, . . . , K K 0,0 ∑ xk = 1 k=1 Mr (xk ) 0, ∀k = 1, . . . , K Mr−1 (θ, xk ) 0, ∀θ ∈ Sk , k = 1, . . . , K This semideﬁnite problem can be solved to obtain the desired upper and lower bounds on the 11 objective function. As r is increased, and information about more moments of X are used in the semideﬁnite program, tighter bounds can be obtained at the cost of increased complexity of the optimization problem. 4 Illustrative Applications In this section, we examine some simple scenarios where the dynamics of the system are properly modeled by the coupled processors. In all the computational results presented below, Gloptipoly [21] and Sedumi [22] were the tools used to solve the semideﬁnite program. 4.1 Modeling Mean User Delays in Cellular Networks 4.1.1 A Two Base Station System Consider a system consisting of two neighboring base stations. A base station serves its associated users according to the FCFS discipline. Each base station serves a user at constant power, or is idle if there are no active users associated with it. We will focus in this case on the interference- limited setting, and examine a scenario where the users in the system are concentrated at two locations. The users who are served by base station 1 are concentrated at a location close to it, such that the received signal strength from base station 1 is very high while any interference from base station 2 is greatly attenuated. These users are immune to interference from base station 2, and can be served at a rate of 4 Mbps both when base station 2 is transmitting and when it is idle. The users associated with base station 2 are close to the cell edge and are served at a rate of 2.5 Mbps when base station 1 is transmitting, and a rate of 4 Mbps when base station 1 is idle. Users arrive at each location according to a Poisson process with rate λ and with service requirements that are exponentially distributed with mean 1 Mb. When a users’ service requirement is met, the user leaves the system. Fig. 1 depicts the bounds on the mean delay obtained by using progressively higher order relaxations, along with the simulated mean delay. The bounds on the delay are obtained by solving the semideﬁnite program with the mean sum queue length as the objective function, and then using Little’s law. The bounds obtained from assuming that the neighboring base station is always 12 1.2 1.1 1 Simulated Mean Delay Upper Bound, r = 1 Lower Bound, r = 1 0.9 Upper Bound, r = 2 Lower Bound, r = 2 0.8 Upper Bound, r = 3 Delay Lower Bound, r = 3 Upper Bound, always busy 0.7 Lower Bound, always idle 0.6 0.5 0.4 0.3 0.5 1 1.5 2 Lambda Figure 1: Mean delay for a two base station system busy/idle are also plotted for comparison. Even when r = 1, the bounds computed through the optimization proposed above are much closer to the actual steady state mean delay. Further, as the relaxation order is increased, the upper and lower bounds are very close, and match the simulated results very well. The distance between the upper bound that assumes the system is saturated and the upper bounds obtained by solving the SDP clearly indicates the impact that coupling has on the mean delay may be more complex than simply assuming persistent interference, even under heavy loads. In this case, closed form expressions for the steady state delay are known, however, the degree of accuracy of our results is highly encouraging. 4.1.2 A Three Base Station System Next, we consider a system with three base stations shown in Fig. 2(a). Since inter-cell interfer- ence is a local phenomenon, and users are not affected much by transmissions at base stations located far away, studying such systems with a small number of base stations can thus provide insight into the performance experienced by users in larger systems. Consider the scenario where users associated with each base station are nearly equidistant from all the base stations as in Fig. 2(a), and receive almost equal signals from each base station. Each base station serves its associated users at a rate of 5 Mbps when they are all busy, and at a rate of 7 Mbps when two of the base stations are busy, and a base station serves users at a rate of 10 Mbps if it is the only busy 13 Base Base station 1 station 2 sector for base station 3 Base station 3 (a) System conﬁguration Base station status Service rates (Mbps) BS 1 BS 2 BS 3 BS 1 BS2 BS3 Idle Idle Idle 0 0 0 Idle Idle Busy 0 0 10 Idle Busy Idle 0 10 0 Idle Busy Busy 0 7 7 Busy Idle Idle 10 0 0 Busy Idle Busy 7 0 7 Busy Busy Idle 7 7 0 Busy Busy Busy 5 5 5 (b) Service Rates Figure 2: Three base station system base station. The state-dependent service rates are summarized in Table 2(b). The mean rate of arrival at each base station is again λ. For this queuing system, no closed form expressions or tight bounds for the steady state delay are available to the best of our knowledge. As shown in Fig. 3, the upper and lower bounds are not far apart, and closely trail the simulated mean delay, and in particular capture the trends with increasing load. The bounds assuming the queues are saturated are much worse, demonstrating that the nature of the coupling plays a signiﬁcant role in the performance experienced by the users in the system. 14 3.5 3 Simulated Mean Delay Upper Bound, r = 1 Lower Bound, r = 1 2.5 Upper Bound, always busy Lower Bound, always idle Delay 2 1.5 1 0.5 1 1.5 2 2.5 3 3.5 4 4.5 Lambda Figure 3: Mean delay of a three base station system. 4.2 A Generalized Processor Sharing System 4.2.1 Two Queue System Consider a server with a service rate of 10 that is shared by two classes of packets with equal weights, i.e., R1 = R2 = 10 and φ1 = φ2 . packets arrive at each class as a Poisson process of rate λ, with exponentially distributed service requirements with mean 1. In this case, since the sum service rate is a constant, independent of the system state, the sum queue length process is a Markov process, and the distribution of the sum queue length(but not the individual queue lengths) can be analytically determined. The bounds computed by solving the SDP with r = 1, 2 are plotted in Fig. 4, along with the simulated results. The lower and upper bounds match exactly, for all the values of λ. In addition, they precisely match the analytically computed mean delay (not shown). This is conﬁrmed by the simulated mean delay matching the bound. Fig. 5 exhibits the mean queue length at the queue associated with one of the classes as λ increases. While the system as a whole is work-conserving, the individual queues are not. The bounds computed by solving the SDP with r = 1, 2 are plotted along with the bounds computed assuming that the neighboring queue is always busy/idle. Note that, in this case closed-form expressions for the mean delay are known [12, 14]. However, for larger systems, analytic expres- sions for the mean delay are not known. 15 2.5 Mean Delay for users entering the system 2 Simulated Mean Delay Upper Bound, r = 1 Lower Bound, r = 1 1.5 Upper Bound, r = 2 Lower Bound, r = 2 1 0.5 1 1.5 2 2.5 3 3.5 4 4.5 Lambda Figure 4: Mean overall system queue length for a 2 Queue GPS system 5 4.5 4 Mean Delay for users in Queue 1 3.5 3 Simulated Mean Delay 2.5 Upper Bound, r = 1 Lower Bound, r = 1 2 Upper Bound, r = 2 Lower Bound, r = 2 1.5 Upper Bound, always busy Lower Bound, always idle 1 0.5 0 0.5 1 1.5 2 2.5 3 3.5 4 4.5 5 Lambda Figure 5: Mean queue length at one of the queues in the two queue GPS system 4.2.2 Three Queue System Consider the case now, where the GPS server is shared by three classes of packets with weights φ1 = 3, φ2 = 1, φ3 = 1. packets arrive at each class as a Poisson process of rate λ, with expo- nentially distributed service requirements with mean 1. The sum queue length process is again a Markov process, and the mean sum queue length is plotted in Fig 6, along with the computed bounds. Figs. 7 and 8 exhibit the mean queue lengths at queue 1 and at queue 2 respectively. The mean delay observed at queue 3 is identical to that observed at queue 2. In this case, note that trying to bound the delay at queues 2 and 3, assuming that the other queues are saturated and queue 1 is always busy leads to a predicted mean delay of ∞. This is because, the system is only stable 16 4 3.5 Mean Delay for a user in the system 3 2.5 Simulated Mean Delay 2 Upper Bound, r = 1 Lower Bound, r = 1 1.5 Upper Bound, r = 2 Lower Bound, r = 2 1 0.5 1 1.5 2 2.5 3 Lambda Figure 6: Mean system queue length Queue: 1 0.35 Simulated Mean Delay Upper Bound, r = 1 Lower Bound, r = 1 Upper Bound, r = 2 Mean Delay for users in Queue 1 0.3 Lower Bound, r = 2 Upper Bound, always busy Lower Bound, always idle 0.25 0.2 0.15 1 1.5 2 2.5 3 Lambda Figure 7: Mean queue length at queue 1 because queues 2 and 3 can borrow the excess capacity of the server when queue 1 is idle. Our simulations demonstrate that the SDP formulation provides reasonably tight bounds on the mean queue lengths of the individual queues when r ≥ 2. 4.2.3 Choosing Queue Weights The choice of class weights is a crucial factor affecting the performance experienced by packets in the system. In systems, such as wireless systems, where the maximum rates at which different classes can be served varies, a systematic method to choose the class weights is not apparent. Note that, in this case, even the sum queue length process is not a Markov process, as the system is non- work conserving. The close bounds provided by our formulation can be exploited to determine 17 Queue: 2 10 9 8 Mean Delay for users in Queue 2 7 6 Simulated Mean Delay Upper Bound, r = 1 5 Lower Bound, r = 1 Upper Bound, r = 2 4 Lower Bound, r = 2 3 Lower Bound, always idle 2 1 1 1.5 2 2.5 3 Lambda Figure 8: Mean queue length at queue 2 the weights that should be associated with each class, if the objective function is a weighted combination of the queue length moments. 8 7 6 Mean Queue Length 5 z=1 Queue: 1 4 z=1 Queue: 2 z=2 Queue: 1 z=2 Queue: 2 3 z=4 Queue: 1 z=4 Queue: 2 2 1 1 1.2 1.4 1.6 1.8 2 2.2 2.4 2.6 2.8 3 Lambda Figure 9: Mean queue lengths resulting from the different objective functions Consider a two queue system, with R1 = 10, R2 = 5, where the arrival rate to each class is equal to λ. We consider an objective function of the form E[Qz + Qz ], and use the lower bound 1 2 from the SDP formulation to approximate the value of the objective function for various class weights. The weight associated with class 1 is assumed to be φ1 = φ, the weight associated with class 2 is then φ2 = 1 − φ. We do a simple line search to determine the weight to be associated with class 1 to minimize the objective function. As z is increased, the penalty associated with high queue lengths increases rapidly, and the objective function tends to balance the performance experienced by the queues. 18 1.1 1 0.9 z=1 z=2 0.8 ’Optimal’ Weight z=4 0.7 0.6 0.5 0.4 1 1.2 1.4 1.6 1.8 2 2.2 2.4 2.6 2.8 3 Lambda Figure 10: Optimal weights for the different objective functions Fig. 9 exhibits the mean queue lengths of the two queues against the arrival rate λ, for z = 1, 2, 4, and Fig. 10 plots the weight determined using the SDP formulation. When z = 1, the objective is to minimize the system queue length. The optimal policy here is to always serve queue 1, whenever both queues are busy and φ = 1 irrespective of the load, as shown in Fig. 10. However, this leads to very large queue lengths at queue 2. Increasing the value of z leads to larger weights being associated with queue 2, in order to balance the queue lengths at the two queues. Fig. 9 clearly shows the impact of the larger weights that are picked using the SDP formulation on the mean queue lengths of the two queues. 4.3 A Multiple Access System The ﬁnal example we consider is an ALOHA-like [23, 24] slotted multiple access system, where N users contend for access to a shared channel in order to communicate with a central server. We use a collision model to capture the effect of simultaneous transmissions, i.e., if a user is the only one contending for the channel in a slot, the transmission is successful and a 1Kb portion of the ﬁle is transmitted to the sever. If more than one user transmits at the same time, there is a collision and all transmissions are unsuccessful. Files with exponentially distributed sizes and mean 2Mb arrive for transmission at each user as a Poisson process with mean arrival rate λ. All users with ﬁles waiting for transmission contend at a time slot with probability PC . Our aim is to ﬁnd a contention probability that results in low ﬁle transfer delays and high user throughputs. 19 Clearly, the contention probability has to be chosen in a load-dependent manner. Intuitively, when the system load is low, the number of users with non-empty queues at any given time will be low and those users should contend with higher probability. However, at high loads, when a larger number of users are likely to have non-empty queues users should contend with a lower probability in order to avoid collisions. If all N users are saturated, the contention probability resulting in the highest throughput is PC = 1/N. This can be seen easily by differentiating the probability of there being exactly one contending user, Pr.(exactly one contending user) = Pr.(successful transmission) = PC (1 − PC )N−1 , with respect to the contention probability PC and equating to zero. However, in the dynamic system, the number of busy users varies as ﬁles arrive and are transmitted. In this case, it is not clear how the contention probability should be chosen. We model this system using a ﬂuid coupled processors model, such that each user transmits data at a rate of 1000PC (1 − PC )n−1 when n of the N users have data to send. Our goal is to ﬁnd, using the lower bound from the semideﬁnite formulation as an approximation, the contention probability that minimizes the mean sum system queue length. We consider a system with three users, and as before use a simple line search to ﬁnd the common contention probability that minimizes the mean sum queue length of the system. 0.8 0.7 ’Optimal’ probability Pr.(contention) = 1/(number of users) Probability of contention 0.6 0.5 0.4 0.3 0.1 0.2 0.3 0.4 0.5 0.6 0.7 Lambda Figure 11: Optimal probability of contention Fig. 11 shows how the contention probability determined using the SDP formulation varies 20 with the ﬁle arrival rate λ. As expected, the contention probability is high when load is low and collisions are rare, and decreases with increasing loads in order to avoid collisions. At very high loads, the contention probability approaches 1 , the optimal contention probability in the saturated 3 system where all users have backlogged queues. 8 7 6 ’Optimal’ probability 5 Pr.(Contention) = 1/3 Mean Delay 4 3 2 1 0.1 0.2 0.3 0.4 0.5 0.6 0.7 Lambda Figure 12: Mean delay comparison 6 x 10 7 6 Throughput (bits/sec.) ’Optimal’ probability 5 Pr.(Contention) = 1/3 4 3 2 1 0.1 0.2 0.3 0.4 0.5 0.6 0.7 Lambda Figure 13: Mean throughput comparison Figs. 12 and 13 exhibit the mean ﬁle transfer delay and mean user throughput for the system using the contention probability determined using our SDP formulation and the system with a contention probability of 1 . At very high loads, performance of the two schemes are close to 3 identical. This is to be expected as the probability of contention chosen using the semideﬁnite formulation is very close to 1 . At lower loads, the mean ﬁle transfer delay and the mean user 3 throughputs are much improved when using the contention probability from the SDP formulation. 21 This is particularly clear from the plot of the mean user throughputs against the user arrival rate. Thus, we can clearly see that the coupled processor model accurately reﬂects the dynamics of the system, and shows the potential of the semideﬁnite formulation in modeling and optimizing such systems. 5 Conclusion and Future Work We proposed an approach to model coupled queuing systems using a semideﬁnite programming approach, and obtained bounds on linear functions of the moments of the queue lengths in the system at steady state. The resultant bounds were used to study some example scenarios, where they were found to closely approximate the actual values. The proposed approach can be ex- tended, to arrive at bounds on the tail probabilities of the individual queue length distributions in a manner similar to that used in [15]. Another possible area of study is to examine systems with general service requirements, or those where the service requirements are modeled using phase type distributions. Another area that requires further study is when the queues in the sys- tem support multiple classes of users. Such a model could potentially be used to better model the dynamics of wireless networks coupled by interference. The various user classes could be used to capture the differences between users at different locations within the network. This would lead to a better understanding of the effect of interference on the delays perceived by users. One drawback of the proposed formulation is the exponential scaling of the size of the semidef- inite optimization as the number of queues increases. This is not surprising, as the size of the problem also increases exponentially with the number of queues. 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